A transitive group $G$ of permutations of a set $Ω$ is primitive if the only $G$-invariant equivalence relations on $Ω$ are the trivial and universal relations.
If $α\in Ω$, then the orbits of the stabiliser $G_α$ on $Ω$ are called the $α$-suborbits of $G$; when $G$ acts transitively the cardinalities of these $α$-suborbits are the subdegrees of $G$.
If $G$ acts primitively on an infinite set $Ω$, and all the suborbits of $G$ are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of $G$ as a non-decreasing sequence $1 = m_0 \leq m_1 \leq ...$, the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture there might exist a number $c$ which perhaps depends upon $G$, perhaps only on $m$, such that $m_r \leq c(m-2)^{r-1}$.
In this paper it is shown that such an enumeration is not desirable, as there exist infinite primitive permutation groups possessing no infinite subdegree, in which two distinct subdegrees are each equal to the cardinality of infinitely many suborbits. The examples used to show this provide several novel methods for constructing infinite primitive graphs.
A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann's question may be answered, at least for groups exhibiting suitable rates of growth.
The subdegrees of a transitive permutation group are the orbit lengths of a point stabilizer. For a finite primitive permutation group which is not cyclic of prime order, the largest subdegree shares a non-trivial common factor with each non-trivial subdegree. On the other hand it is possible for non-trivial subdegrees of primitive groups to be coprime, a famous example being the rank 5 action of the small Janko group on 266 points which has subdegrees of lengths 11 and 12. We prove that, for every finite primitive group, the maximal size of a set of pairwise coprime non-trivial subdegrees is …
If $G$ is a group of permutations of a set $Ω$ and $α\in Ω$, then the {\em $α$-suborbits} of $G$ are the orbits of the stabilizer $G_α$ on $Ω$. The cardinality of an $α$-suborbit is called a {\em subdegree} of $G$. If the only $G$-invariant equivalence classes on $Ω$ are the trivial and universal relations, then $G$ is said to be a {\em primitive} group of permutations of $Ω$. In this paper we determine the structure of all primitive permutation groups whose subdegrees are bounded above by a finite cardinal number.
Dolfi, Guralnick, Praeger and Spiga asked if there exist infinitely many primitive groups of twisted wreath type with nontrivial coprime subdegrees. Here we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees. In particular, we define a primitive twisted wreath group $G(m,q)$ constructed from the nonabelian simple group $\text{PSL}(2,q)$ and a primitive permutation group of diagonal type with socle $\text{PSL}(2,q)^m$, and determine all values of $m$ and $q$ for which $G(m,q)$ has nontrivial c…
In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H acting completely reducibly on a vector space V: if the orbits containing the vectors a and b have coprime lengths m and n, we prove that the orbit containing a+b has length mn. Such groups H are always reducible if n and m are greater than 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor. In the more general context of finite primitive permutation groups G, we show that coprime non-identity sub…
If $G$ is a group of permutations of a set $Ω$, then the suborbits of $G$ are the orbits of point-stabilisers $G_α$ acting on $Ω$. The cardinalities of these suborbits are the subdegrees of $G$. Every infinite primitive permutation group $G$ with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph $Γ$ with vertex set $Ω$, and there is consequently a natural action of $G$ on the ends of $Γ$. We show that if $G$ is closed in the permutation topology of pointwise convergence, then the structure of $G$ is determined by the len…
This paper is about the structure of infinite primitive permutation groups and totally disconnected locally compact groups ("tdlc groups'"). The permutation groups we investigate are subdegree-finite (i.e. all orbits of point stabilisers are finite). Automorphism groups of connected, locally finite graphs are examples of subdegree-finite permutation groups. The tdlc groups we investigate all have a maximal subgroup that is compact and open. Tdlc groups with few open subgroups (recently studied by Pierre-Emmanuel Caprace and Timothée Marquis) are examples of such groups. We prove a classificati…
A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less obvious ones. We prove that 3/2-transitive groups are either affine or almost simple, and classify the latter. One of the main steps in the proof is an arithmetic result on the subdegrees of groups of Lie type in characteristic $p$: with some explicitly listed exceptions, every primitive action of such a group is either 2-transitive, or has a subdegree divisible …
We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also classify finite almost simple groups that have a maximal subgroup isomorphic to $\mathrm{Alt}(5)$ or $\mathrm{Sym}(5)$.
The problem of describing the invariance groups of unordered relations, called briefly \emph{relation groups}, goes back to classical work by H. Wielandt. In general, the problem turned out to be hard, and so far it has been settled only for a few special classes of permutation groups. The problem have been solved, in particular, for the class of primitive permutation groups, using the classification of finite simple groups and other deep results of permutation group theory. In this paper we show that, if $G$ is a finite simple primitive permutation group other then the alternating group $A_n$…
In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and…
Let $G$ be a permutation group acting on a set $V$. A partition $π$ of $V$ is distinguishing if the only element of $G$ that fixes each cell of $π$ is the identity. The distinguishing number of $G$ is the minimum number of cells in a distinguishing partition. We prove that if $G$ is a primitive permutation group and $|V|\ge336$, its distinguishing number is two.
Let $G$ be a finite permutation group on $Ω$. An ordered sequence of elements of $Ω$, $(ω_1,\dots, ω_t)$, is an irredundant base for $G$ if the pointwise stabilizer $G_{(ω_1,\dots, ω_t)}$ is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same size we say that $G$ is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost simple, of affine-type, or of diagonal type. Moreover we prove that a diagonal-type primitive permutation groups is IBIS if and only if it is isomorphic to $PSL(2,…
This paper presents an analysis of primitive permutation groups of degree $3p$, where $p$ is a prime number, analogous to H. Wielandt's treatment of groups of degree $2p$. It is also intended as an example of the systematic use of combinatorial methods as surveyed in \S6 for distilling information about a permutation group from knowledge of the decomposition of its character. The work is organised into three parts. Part I contains the lesser half of the calculation, the determination of the decomposition of the permutation character. Part II contains a survey of the combinatorial methods and, …
We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}Γ\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.
We conjecture that if $G$ is a finite primitive group and if $g$ is an element of $G$, then either the element $g$ has a cycle of length equal to its order, or for some $r,m$ and $k$, the group $G\leq S_m\wr S_r$, preserving a product structure of $r$ direct copies of the natural action of $S_m$ or $A_m$ on $k$-sets. In this paper we reduce this conjecture to the case that $G$ is an almost simple group with socle a classical group.
We classify the finite primitive groups containing a permutation with at most four cycles (including fixed points) in its disjoint cycle representation.
If $G$ is a group acting on a set $Ω$ and $α, β\in Ω$, the digraph whose vertex set is $Ω$ and whose arc set is the orbit $(α, β)^G$ is called an {\em orbital digraph} of $G$. Each orbit of the stabiliser $G_α$ acting on $Ω$ is called a {\it suborbit} of $G$. A digraph is {\em locally finite} if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph $Γ$ has more than one end if there exists a finite set of vertices $X$ such that the induced digraph $Γ\setminus X$ contains at least two infinite connected components; if there exists such a set containing precis…
Let $G$ be a permutation group on a set $Ω$ of size $t$. We say that $Λ\subseteqΩ$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $Λ$. We define the height of $G$ to be the maximum size of an independent set, and we denote this quantity $\mathrm{H}(G)$. In this paper we study $\mathrm{H}(G)$ for the case when $G$ is primitive. Our main result asserts that either $\mathrm{H}(G)< 9\log t$, or else $G$ is in a particular well-studied family (the "primitive large--base groups"). An immediate corollary of this result is a character…
We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to all primitive permutation groups with finite point stabilizers.
We prove that there exists a universal constant $c$ such that any finite primitive permutation group of degree $n$ with a non-trivial point stabilizer is a product of no more than $c\log n$ point stabilizers.
General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer $ H \leq \mathrm{GL}(V) $, where the vector space $ V $ has dimension $ d $ over the prime field, can be bounded in terms of $ d $ and $ \log |V| / \log |H| $ only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.
Motivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher-O'Nan-Scott theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.
In this paper, we study intersecting sets in primitive and quasiprimitive permutation groups. Let $G \leqslant \mathrm{Sym}(Ω)$ be a transitive permutation group, and ${S}$ an intersecting set. Previous results show that if $G$ is either 2-transitive or a Frobenius group, then $|{S}|\leqslant|G_ω|$ (for some $ω\in Ω$). Furthermore, for some 2-transitive groups, $|{S}|=|G_ω|$ if and only if ${S}$ is a coset of a stabilizer. In this paper, we prove that these statements are far from the truth for general transitive groups. In particular, we show that in the case of primitive groups, there is eve…
Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$. We prove that either $k(G)<n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$, or $G$ belongs to explicit families of examples.
We investigate the proportion of fixed point free permutations (derangements) in finite transitive permutation groups. This article is the first in a series where we prove a conjecture of Shalev that the proportion of such elements is bounded away from zero for a simple finite group. In fact, there are much stronger results. This article focuses on finite Chevalley groups of bounded rank. We also discuss derangements in algebraic groups and in more general primitive groups. These results have applications in questions about probabilistic generation of finite simple groups and maps between vari…
In January 1969, Peter M. Neumann wrote a paper entitled "Primitive permutation groups of degree 3p". The main theorem placed restrictions on the parameters of a primitive but not 2-transitive permutation group of degree three times a prime. The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree. However, there are further reasons for being interested in this paper. First, it was written at a time when combinatorial techniques were being …
The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primiti…
Working in a theory with an integer-valued dimension on interpretable sets, we classify pseudofinite definably primitive permutation groups acting on one-dimensional sets which satisfy a version of chain condition on centralizers and on point-wise stabilizers. This generalises the classification of pseudofinite definably primitive permutation groups in supersimple theories of finite rank to supersimple theories of infinite rank.
Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.
We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds on the composition length of a finite completely reducible linear group in terms of some of its parameters. In almost all cases we show that the bounds are sharp, and describe the extremal examples.
The primitive finite permutation groups containing a cycle are classified. Of these, only the alternating and symmetric groups contain a cycle fixing at least three points. The contributions of Jordan and Marggraff to this topic are briefly discussed.
Let $G$ be a finite primitive permutation group and let $κ(G)$ be the number of conjugacy classes of derangements in $G$. By a classical theorem of Jordan, $κ(G) \geqslant 1$. In this paper we classify the groups $G$ with $κ(G)=1$, and we use this to obtain new results on the structure of finite groups with an irreducible complex character that vanishes on a unique conjugacy class. We also obtain detailed structural information on the groups with $κ(G)=2$, including a complete classification for almost simple groups.
We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most m(T). Moreover, apart from an explicit list of groups, the bound can be reduced to m(T)/4. These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4.
Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note we state explicit formulae for the primitive idempotents of K\otimes pp_k(G), where pp_k(G) is the ring of p-permutation kG-modules.
Suppose that $G$ is a finite, transitive, solvable permutation group acting on a set $S$ with $n$ elements. Let $G_0$ be the stabilizer of a point $α\in Ω$. Define the rank of a permutation group, denoted $r(G),$ as the number of distinct orbits of $G_0$ in $S$ (including the trivial orbit $\{α\}$). Huppert \cite{Huppert} and Foulser \cite{Foulser} classified all finite, solvable, permutation groups of rank two and three respectively, and Foulser restricted the rank four groups to a small list of possibilities. This paper completes the classification of all groups of rank less than $5$ by expl…
We show that given generators for subgroups $G$ and $H$ of $\mathrm{S}_n$, if $G$ is primitive then generators for $\mathrm{N}_H(G)$ may be computed in quasipolynomial time, namely $2^{O(\log^3 n)}$. The previous best known bound was simply exponential.
A finite non-regular primitive permutation group $G$ is extremely primitive if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, finding various applications. The classification of extremely primitive groups was recently completed by Burness and Lee, who relied on an earlier classification of soluble extremely primitive groups by Mann, Praeger and Seress. Unfortunately, there is an inaccuracy in the latter classification. We correct this mistake, and also investigate regular linear spaces which admit groups of automorphism…
A permutation group is {\it binary} if its orbits on $k$-tuples, for any integer $k\geq 2$, can be deduced from its orbits on $2$-tuples. Cherlin conjectured that a finite primitive binary permutation group $G$ must lie in one of three known families. In this paper we complete the proof of this conjecture. To do this we study the case where the group $G$ is almost simple of Lie type.
We show that the minimal base size $b(G)$ of a finite primitive permutation group $G$ of degree $n$ is at most $2 (\log |G|/\log n) + 24$. This bound is asymptotically best possible since there exists a sequence of primitive permutation groups $G$ of degrees $n$ such that $b(G) = \lfloor 2 (\log |G|/\log n) \rceil - 2$ and $b(G)$ is unbounded. As a corollary we show that a primitive permutation group of degree $n$ that does not contain the alternating group $\mathrm{Alt}(n)$ has a base of size at most $\max\{\sqrt{n} , \ 25\}$.
Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-grou…
Let G be a permutation group, acting on a set Ωof size n. A subset B of Ωis a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r \geq 1 such that Alt(m)^r \unlhd G \leq Sym(m) \wr Sym(r), where the action of Sym(m) is on k-element subsets of {1,...,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M24 in its natural action on 24 points, or b(G) \leq \lceil \log n\rceil+1. F…
Let $G$ be a transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G| < n$ if $G$ is primitive unless $n = 3^{4}$, $5^4$, $3^8$, $5^8$, or $3^{16}$. This bound is sharp when $n$ is prime. In fact, when $G$ is primitive, $|\mathrm{Out}(G)| < n$ unless $G$ is a member of a given infinite sequence of primitive groups and $n$ is different from the previously listed integers. Many other results of this flavor are established not o…
The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of…
We prove that if $G$ is a finite primitive permutation group and if $g$ is an element of $G$, then either $g$ has a cycle of length equal to its order, or for some $r$, $m$ and $k$, the group $G \leq \mathrm{Sym}(m) \textrm{wr} \mathrm{Sym}(r)$ preserves the product structure of $r$ direct copies of the natural action of $\mathrm{Sym}(m)$ on $k$-sets. This gives an answer to a question of Siemons and Zalesski and a solution to a conjecture of Giudici, Praeger and the second author.
We introduce a new product for permutation groups. It takes as input two permutation groups, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M. Under mild conditions on M and N the group M [X] N is simple. As a permutational product, its most significant property is the following: M [X] N is primitive if and only if M is primitive but not regular, and N is transitive. Despite this remarkable similarity with the wreath product in product action, M [X] N and M Wr N are thoroughly dissimilar. The product provides a general way to build exotic …
Let $G$ be a permutation group on a set $Ω$. A subset of $Ω$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $Σ(G)$, which we call the Saxl graph of $G$. The vertices of $Σ(G)$ are the points of $Ω$, and two vertices are adjacent if they form a base for $G$. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of $Σ(G)$ for a finite transitive group $G$, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For in…
Let $G$ be a finite permutation group on $Ω$. An ordered sequence $(ω_1,\ldots,ω_\ell)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups $G$ to the case that the socle of $G$ is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socl…
We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the $2$-homogeneous and the primitive groups. Lastly, we define $k$-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a $k$-reachable permutation group of degree $n$ with $6 \le k \le n - 6$ is eit…
For a finite group $G$, the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ is the space of genus $g$ covers of the Riemann sphere with $r$ branch points and the monodromy group $G$. In this paper, we give a complete list of primitive genus one systems of affine type. That is, we assume that $G$ is a primitive group of affine type. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in $\mathcal{H}^{in}_{r,1}(G)$. Furthermore, we give a new algorithm for computing large braid orbits on Nielsen classes. This algorith…
In the first paper in this series we estimated the probability that a random permutation $π\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $π$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\dots,k_m$, where $k_1+\cdots+k_m=n$. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than $\mathcal{S}_n$ or $\mathcal{A}_n$. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estim…
